Geometry: Non-Euclidean Geometry

Non-Euclidean Geometry

Geometry

Euclid had a hard time with the Parallel Postulate. As has already been mentioned, it is similar enough to the theorem about the existence and uniqueness of perpendicular lines to make a person think that the Parallel Postulate can be proven. Many brilliant mathematicians tried to prove the Parallel Postulate from Euclid's other postulates, and all have failed. It might be comforting to note that their failure was not a reflection of their ability as mathematicians. They were trying to do the impossible. Not just the impossible for their time, but the impossible for all time.

Originally non-Euclidean geometry included only the geometries that contradicted Euclid's 5th Postulate. But then mathematicians realized that if interesting things happen when Euclid's 5th Postulate is tossed out, maybe interesting things happen if other postulates are contradicted. Each time a postulate was contradicted, a new non-Euclidean geometry was created. So the notion of non-Euclidean geometry had to be expanded. A non-Euclidean geometry is a geometry characterized by at least one contradiction of a Euclidean geometry postulate.

Tangent Line

There are several instances where mathematicians have proven that it is impossible to prove something. Although this concept might be difficult to understand and accept, it can be interpreted as permission to stop wasting time trying to prove a particular theorem.

One of the reasons why non-Euclidean geometry is difficult to accept is that it goes against our practical experience. We perceive our world to be flat, even though the earth is spherical. It is easy to visualize a city as a grid with nonintersecting straight streets. That perception works because the curvature of the earth is insignificant when compared to the size of our cities. But non-Euclidean geometry has applications both in space and on our home planet.

Before we leave Euclid's world, it might be wise to remind yourself of the Parallel Postulate.

The Parallel Postulate. Through a given point, not on a given line, only one parallel can be drawn to the given line.

You are at a point in the text when I need to be honest with you. Euclid is credited with being the father of geometry, but geometry has come a long way since Euclid's day. When you read current geometry books (like this one) it is easy to forget that Euclid wrote in Greek, using the language of his time. Although his writings might have been hip in his day, they lose a lot in the translation. There's nothing wrong with that. His writings served their purpose. The ideas he introduced in geometry have furthered development in many fields outside of mathematics, and geometry continues to develop even as I write. The point I am trying to make is that the wording of the definitions, theorems, and postulates in geometry has also changed with time, but its meaning has not. The phrasings of the definitions, theorems, and postulates in this section are equivalent to the ones that Euclid stated years ago, though they are not identical. I am taking a long time to confess my sin. Although I have credited this postulate to Euclid, the phrasing of it really belongs to John Playfair. It is equivalent to the one that Euclid came up with, but it is much more understandable.